Yeah, as long as you have one dedicated CPU core to folding, that will be sufficient. So, a dual core is all you need (one core for everything else, one to feed the GPU). You can get an idea of CPU efficiencies by looking at their TDPs (Thermal Design Power). Typically, power hungry gaming CPUs come in at 95-150 Watts. Basic desktop CPUs have TDPs around 65 watts. Low-power parts can get as low as 25 Watts. Your old i5 had a TDP of 84 watts, which suggests there is some headroom to make a more efficient system if you get a new power efficient dual core.
For putting together something on the cheap, check out processors like this (the link should provide you with a Newegg search showing dual-core processors with TDPs less than 35 watts).
https://www.newegg.com/p/pl?N=100007671 ... 0600039206
Going with a micro-ATX board will also keep background power consumption down a bit. 4 Gigs of RAM will be plenty (just need to run the OS + folding). I've folded on as little as 1 GB in Linux, 2 in windows. 4 is good though and it gives you some headroom.
You can pick up a really cheap 64 GB SSD (used on eBay for like $20) that will also sip power. For a power supply, try to get one with an efficiency rating of 80+ Gold or better. I recommend this:
https://www.newegg.com/seasonic-focus-s ... -_-Product
I got one of those used on eBay for $40 shipped and it cut about 20 watts of power consumption out, making my dedicated folding rig that much more efficient. Note that if you plan on adding another GPU in the future, you'd need a bigger PSU (and perhaps a full size ATX main board).
It is though, much younger than the motherboard, this is why I was wondering if it would make sense to fit it on.